Optimal Accumulation of Human and Physical Capital

in A Model with Consumption Externalities and Altruism†

Been-Lon Chen‡

Academia Sinica

Yu-Shan Hsu# National Chung Cheng University

Chih-Fang Lai*

Soochow University

June 2018

Abstract

This paper studies suboptimality of education and saving decisions in an overlapping-generation model of consumption externalities with parental altruism. Parental altruism is an important human behavior, but its role has neglected by the existing related literature. We find that parents pay education but may not leave bequests for children. Under inoperative bequest motives, equilibrium physical capital is over-accumulated, while human capital is under-accumulated. Thus, implementing public pensions and public education at the same time is called for, which is in line with the current policies employed in OECD. Optimal tax policies are envisaged in order for the equilibrium to attain the first best solution. Keywords: Consumption externalities, Human capital, Intergenerational transfers, Bequests, Pension, Education subsidy. JEL classification: E21; H52 ____________________

†Earlier versions have benefitted from discussions with participants at several seminars. ‡Corresponding address: Been-Lon Chen, the Institute of Economics, Academia Sinica, 128 Academia Road Section 2, Taipei 11529, TAIWAN; Phone: (886-2)27822791 ext. 309; Fax: (886-2)27853946; email: bchen@econ.sinica.edu.tw #Department of Economics, National Chung Cheng University, 168 University Road, Min-Hsiung, Chia-Yi 62102, TAIWAN. Phone: (886-5) 2720411 ext. 34127; Fax: (886-5) 2720816; email: ecdysh@ccu.edu.tw. *Department of Economics, Soochow University, 56 Kuei-Yang Street Section 1, Taipei 10048, TAIWAN; Phone: (886-2)23111531 ext. 3582; Fax: (886-2)23822001; email: cflai@scu.edu.tw.

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1. Introduction

This paper studies the efficiency issue of capital accumulation in an overlapping generations (OLG)

economy with both physical and human capital. Since Diamond (1965), it has been well-known that

physical capital accumulation path needs not be dynamically efficient. In the case of over-accumulation,

intergenerational arrangements such as pensions might be useful in order to increase the welfare of all

generations. Recent literature has stressed the importance of human capital on economic growth, but

when human capital is taken into account, the problem becomes complicated.1 Compared to the social

optimum, in equilibrium both types of capital may under- or over-accumulate, or one type under-

accumulates and the other type over-accumulates.

The issue has been studied by Boldrin and Montes (2005) and Docquier et al. (2007). In a three-

period OLG model where agents borrow to invest in education,2 Boldrin and Montes (2005) studied

suboptimality of both education and saving decisions. Because of a missing credit market for human

capital investment, these authors found that the long-run competitive equilibrium features under-

accumulated human capital along with over-accumulated physical capital. Their paper posits a possibility

wherein the levels of accumulation of physical and human capital differ from the social optimum in

opposite directions. Later, Docquier et al. (2007) stress the role of intergenerational human capital

externalities. These authors added these externalities into the model of Boldrin and Montes (2005) but

allowed for a perfect credit market for human capital investment. They compared the competitive

equilibrium with that of the social optimum, wherein the social planner’s objective function is the sum

of the lifetime utilities of agents over generations discounted by a social weight. For proper values of the

social weight, they uncovered that the long-run equilibrium undergoes under-accumulation of human

capital along with over-accumulation of physical capital, like that in Boldrin and Montes (2005). Thus, in

these two pieces of work, the combination of education subsidies (like public education) and transfers

from the young to the old (like public pensions) can restore efficiency and increase the welfare.

Recently, Bishnu (2013) has contended a lack of empirical backing for human capital externalities and

instead advocated the importance of consumption externalities.3 He revisited the Docquier et al. (2007)

model by dropping human capital externalities and adding consumption externalities. Under this setup,

he uncovered that physical and human capital either both under- or both over-accumulate relative to the

1 For theoretical literature, see Lucas (1988) and Romer (1990), and for empirical work, Levine and Renelt (1992) and Barro and Lee (1993). 2 Agents pay for their education when young, and work, consume and save in the middle-aged, and retire when old. 3 Consumption externalities were emphasized in early work by Veblen (1912) and were first formalized as a determinant of aggregate consumption by Duesenberry (1949) in his development of the relative income hypothesis. Empirical studies have provided support for the significance of consumption externalities associated with social comparison; see, for example, Clark and Oswald (1996) and Luttmer (2005)

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social optimum. With these results, Bishnu (2013) suggested that implementation of public pensions and

public education at the same time is never recommendable. However, the recommendation is at odd with

what we observe in real world. In OECD, countries always employ the two policies of public pensions

and public education at the same time.4 Such inconsistency motivates a general question: how can we

justify such public intervention in education and pensions when consumption externalities are prevailing?

The purpose of this paper is to show that that parental altruism helps rationalize public involvement

in pensions and education subsidies at the same time. Specifically, we study suboptimality of education

and saving decisions in an otherwise Bishnu (2013) model except for altruism. In our model, parents have

motives to leave bequests and finance education for their children. While leaving bequests for children is

a parental transfer in goods, financing education for children is a parental transfer in kind. Both are

parental altruism toward children. Parental altruism is an important human behavior, but its role has not

been considered in Boldrin and Montes (2005), Docquier et al. (2007) and Bishnu (2013).

Parental bequest motives constitute an important channel for decisions on savings and capital

accumulation in OLG models.5 The literature dates back the early contribution by Barro (1974) in an

OLG model, who showed that bequest motives lead to the Ricardian equivalence of the debt neutrality.

Later, Buiter (1979) and Carmichael (1982) found that the debt neutrality proposition hinges on whether

or not bequest motives are operative. Weil (1987) formally determined a necessary and sufficient

condition for bequest motives to be operative or inoperative. Abel (1988) analyzed the effect of fiscal

policy on the capital evolution under operative and inoperative bequest motives.6

Regarding parental finance in children’s education, many existing studies find that parents are willing

to pay for children’s high school education and beyond (Steelman and Powell, 1991). Surveys carried out

by the Sallie Mae Education Institute disclose that financing children’s college education is among parents’

most important investment they can make (Miller, 1997).7 Using Consumer Expenditure Survey over

the period of 1972-2007, Kornrich and Furstenberg (2007) find that about 50% of parental spending per

child goes to education. Moreover, not just parents in developed countries, parents in developing

4 For pensions in OECD, readers are referred to http://www.oecd.org/els/public-pensions, and for public spending on education, referred to https://data.oecd.org/eduresource/public-spending-on-education.htm. 5 See Michel et al. (2006) for a survey about bequest motives. 6 Other than pure altruism used in Barro (1974), several alternative bequest motives have been proposed, including incomplete annuity markets (Abel, 1985), strategic bequest behavior (Bernheim et al., 1985), joy of giving (Abel, 1988) and warm-glow giving (Andreoni, 1990). 7 According to surveys conducted by Sallie Mae commissioned Gallup & Robinson, Inc. and reported by Miller (1997), most parents (92%) surveyed agree with the statement, “A college education is among the most important investment I will make for my child,” and “even with what it costs today, college is still a good investment.” The importance placed on their children’s education is confirmed, when 31% of the parents report that children’s college education is their highest financial priority, second only to the 38% of parents who indicate everyday budget is their first financial priority and almost twice the 17% of parents who name retirement as their first priority.

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countries are also willing to finance children’s education. For example, using 1985-1986 Peru living

Standard Survey, Gertler and Glewwe (1990) reveal that parents in Peru, and even in rural areas, are

willing to pay for children’s education.8

Recently, several papers have considered altruism when analyzing various issues.9 Closest to our

paper is Alonso-Carrera et al. (2008), who analyzed suboptimal allocation of physical capital in an OLG

model with consumption externalities and altruism. Our paper is different from theirs, because we also

analyze the allocation of human capital arising from consumption externalities and altruism. Our paper

may be thought of as extending Alonso-Carrera et al. (2008) to one with human capital investment. We

envisage the role of bequest motives on whether the accumulation of physical and human capitals differs

from the social optimum in opposite directions, an issue that cannot analyze in Alonso-Carrera et al. (2008).

We find that physical and human capital are under- or over-accumulated depends on whether or not

the bequest motive is operative. When the bequest motive is operative, there is either over-accumulation

or under-accumulation of both physical and human capitals, which is the same as that in Bishnu (2013),

with the levels of accumulation of both types of capital different from the social optimum in the same

direction. By contrast, when the bequest motive is inoperative, there is over-accumulation of physical

capital and under-accumulation of human capital. Thus, the levels of accumulation of physical and human

capitals differ from the social optimum in opposite directions, which is different from Bishnu (2013).

We should remark that our results of opposite accumulation directions are similar to Docquier et al.

(2007), but they arise from different mechanisms. While our mechanism is an inoperative bequest motive,

the mechanism in Docquier et al. (2007) is an intergenerational human capital externality. We should also

comment that the suboptimality in Bishnu (2013) is based on comparisons between the Diamond (1965)

model and the corresponding planner problem. His analysis does not allow one to distinguish the

deviation of optimality due to consumption spillovers from the suboptimality due to bequest motives. In

contrast, our suboptimality is based upon the comparison between the Barro (1974) model and the

corresponding planner problem. The mechanism leading to the lack of optimality in our model depends

not only on consumption externalities but also on bequest motives. When the bequest motive is operative,

consumption externalities are the only source of suboptimality. However, when the bequest motive is

8 In estimates made by Gertler and Glewwe (1990), it was found that rural Peruvian households at all income levels are willing to pay fees high enough to cover the operating costs of new schools in their villages. See similar results found by Lillard and Willis (1997) for Malaysia and by Brown (2006) for China. 9 Page (2003) empirically tested and found the existence of an operative bequest motive. Alonso-Carrera et al. (2007) examined how consumption habits and aspirations make the willingness to leave bequests stronger or weaker. Barnett et al. (2013) investigated how bequest-giving behavior give rise to deviant generations, generations that do not leave a bequest having received an inheritance, and vice versa. Ihori et al. (2017) analyzed a model with altruism and liquidity constraints wherein children’s education costs are shared between children and parents with the share determined by parents.

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inoperative, the source of suboptimality comes from both consumption externalities and inoperative

bequest motives. Our analysis shows how the suboptimality initiating from consumption externalities

depends on inoperative bequest motives.

Finally, we investigate how the tax policy can be used to restore efficiency and increase the social

welfare. We find that capital income is taxed or subsidized, depending on the externality factor being

smaller or larger than unity.10 Moreover, education expenditure is subsidized or taxed, depending on the

altruism factor being smaller or larger than the social weight. Only under operative bequest motives are

bequests taxed or subsidized, and they are taxed if the altruism-factor-discounted externality factor is

larger than the social weight, and subsidized if otherwise. By contrast, under inoperative bequest motives,

there is no bequest in equilibrium, but the optimal bequest is negative. Then, the social optimum calls for

intergenerational lump-sum transfers from the working-aged to retirees if education is subsidized or taxed

at an amount smaller than the absolute value of optimal bequests, and calls for intergenerational lump-

sum transfers from the retirees to the working-aged if otherwise.

We organize this paper as follows. In Section 2, we set up an OLG model with altruism, with the

competitive equilibrium and the social optimum analyzed in Subsections 2.1 and 2.2, respectively. In

Section 3, we compare the competitive equilibrium with the social optimum. Section 4 analyzes optimal

policies to restore optimality. Finally, we offer some concluding remarks in Section 5.

2. The model

Our model follows from Boldrin and Montes (2005), Docquier et al. (2007) and Bishnu (2013). The

economy comprises a sequence of overlapping generations, an initial old generation, and an infinitely

lived government. Agents live for three periods of life: young, middle-aged (or working-aged), and old.

Specifically, our model is otherwise the same as Bishnu (2013) except for allowing for altruism wherein

parents may finance young children’s education and the old may leave bequests for children. In the first

period of life, agents do not consume and they receive education. In the second period of life, agents

work, consume, save and pay children’s education. In the final period of life, agents retire, and consume

and leave bequests for children.

The generation that works in period t is indexed by t. Thus, agents born in period t-1 are referred

to as generation t. Let the population of generation t be Nt which grows at the rate of n, i.e., Nt = nNt-1.

Then, Nt-1 +Nt+Nt+1 is the total population size in period t. Denote by et-1 the education cost of an agent

born in period t-1 (i.e., a generation t agent), which is paid by parents of generation t-1. The education

10 While the externality factor will be defined later, it suffices to note that if the externality factor is larger than unity, then consumption externalities from the middle-aged are larger than those from the old; and vice versa.

5

investment results in a level of human capital in period t as described by the following process

1( ),t th e (1)

where ϕ(e) is a positive, strictly increasing and strictly concave function, that satisfies the Inada condition,

namely, ϕ (́0)=∞ and ϕ (́∞)=0. Notice that, like Bishnu (2013) and different from Docquier et al. (2007),

the human capital accumulation is free from any effect of externality.

The economy has a final good Yt, produced by a neoclassical technology ( , ),t t tY F K H where Kt

is aggregate physical capital and Ht is aggregate human capital, with F assumed to be homogeneous of

degree 1 and satisfying the Inada condition. Denote capital per capita and human capital per capita at the

beginning of period t by t

t

K

t Nk and ,t

t

H

t Nh respectively. Moreover, denote the ratio of physical

capital per capita to human capital per capita at the beginning of period t by ,t t

t t

k K

t h Hk which is the

economy’s average capital per unit of human capital. For simplicity, k is referred to as effective capital

per capita. Then, we can rewrite ( ),t t tY H f k where ( ) 0f and ( ) 0 ( ).f f For simplicity, we

assume that physical capital depreciates completely in one period. Assuming that the factor markets are

competitive, then in optimum, firms rent capital and hire labor to levels that equalize their factor price to

their marginal product.

( ) ( ), ( ) 0,t t t tR R k f k R k (2a)

( ) ( ) ( ), ( ) 0,t t t t t tw w k f k k f k w k (2b)

where Rt is the gross return to capital in period t and wt is the wage rate of raw labor in period t.

Let ct and dt+1 denote the consumption of a generation t agent when middle-aged and old,

respectively. As in Abel (2005), Alonso-Carrera et al. (2008) and Bishnu (2013), an agent obtains utility

from effective consumption that compares her own consumption with a consumption reference. The

felicity function is 1

ˆˆ( , ),t tc du where

t̂c and 1

ˆtd

are an agent’s effective consumption when middle-

aged and old, respectively. The felicity function is strictly increasing and strictly concave, is additive in its

two arguments, and satisfies the Inada condition. Effective consumption is given by ˆ ( , )t t t tc c c d

and 1 1 1 1

ˆ ( , ),t t t td d c d where consumption references ( , )t tc d and 1 1( , )t tc d

represent

negative external effects caused by average consumption jc and jd of relevant generations j. As the

existing papers, we assume that ( , )t tc d and 1 1( , )t tc d are linear with respect to ,t tc d , 1tc and

1,td with partial derivatives 1

, ,t t tc d c

and

1td all being smaller than one. Note that

tc and

1td measure the degree to which a middle-aged agent and an old agent, respectively, keeps up with the

consumption of agents in the same generation. Similarly, td and

1tc gauge the degree to which a

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middle-aged agent and an old agent keeps up with the consumption of agents in the other generation.

Let 1( , )t tV e b

denote the lifetime utility of a generation t agent, who receives education at a cost

et-1 when young and inherits an amount of bequests bt when middle-aged. A parent is altruistic towards

children. Following Alonso-Carrera et al. (2008), the lifetime utility of a generation t agent is

1 1 1ˆˆ( , ) max ( , ) ( , ) ,t t t t t tV e b u c d V e b

(3)

where 1( , )t tV e b

is the lifetime utility of generation t+1, and [0,1) is the altruism factor.

A working-aged agent distributes labor income wtht and inheritance bt among consumption ct,

savings st and children’s education et. Moreover, an old individual receives the return from savings and

allocates it between consumption dt+1 and bequests for children bt+1. Their budget constraints are,

respectively,

,t t t t t tw h b c s ne (4a)

1 1 1.t t t tR s d nb (4b)

The optimization problem of the representative generation t agent is to maximize (3) with respect

to 1 0, , , ,t t t t t

c s e b

subject to (1), (4a) and (4b) with bt+10, given k0 and h0 and taking as given wt and

Rt+1 for all t. The optimization conditions are as follows. First, by the envelope theorem, we obtain

ˆ 1 1

1

ˆˆ( , ) ( ),t

tc t t t t

t

Vu c d w e

e

(5a)

ˆ 1ˆˆ( , ).

t

tc t t

t

Vu c d

b

(5b)

Moreover, the optimality conditions for st, et and bt+1, along with the use of (5a) and (5b), give

1ˆˆ 1 1 1

ˆ ˆˆ ˆ( , ) ( , ) ,t t

c t t t t tdu c d u c d R

(6a)

1

1ˆ ˆ1 1 2 1

ˆ ˆˆ ˆ( , ) ( , ) ( ),t t

t

c t t c t t t t

t

Vnu c d u c d w e

e

(6b)

11

1ˆ ˆ1 1 2

1

ˆ ˆˆ ˆ( , ) ( , ),tt

t

t t c t tdt

Vnu c d u c d

b

(6c)

where the equality in (6b) and (6c) holds if et>0 and bt+1>0, respectively.

Condition (6a) is the Euler equation for consumption between middle and old ages. Condition (6b)

is the decision for paying children’s education. As ϕ(e) satisfies the Inada condition, (6b) holds with an

equality and thus, et is positive. Hence, a parent invests in children’s education to the level, when the loss

in the marginal utility of consumption equals the gain in the (altruism-factor) discounted future marginal

lifetime utility arising from children’s marginal utility of consumption due to higher wage income.

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Condition (6c) determines optimal bequests. If the bequest motive is operative, (6c) holds with an

equality and bt>0. Then, a parent leaves bequests to children to the level when the decrease in her marginal

utility of consumption equals the increase in the (altruism-factor) discounted future marginal lifetime

utility arising from children’s marginal utility of consumption. By contrast, if the bequest motive is not

operative, (6c) holds with an inequality and then, bt=0.

In a symmetric equilibrium, 1 1, ,t t t t t tc c d d c c and

1 1.t td d Moreover, with capital

depreciating fully in one period, aggregate savings in period t are the capital stock at the beginning of

period t+1. Thus, the goods market clearing condition is

1 1 1.t t t ts nk nk h (7)

2.1 Competitive equilibrium

This subsection analyzes the competitive equilibrium.

Definition 1. For given h0 and k0, a competitive equilibrium is the path 0

, , , , , , , ,t t t t t t t t t tc d k s e b h w R

that satisfies firms’ optimization conditions (2a)-(2b), agents’ optimization conditions (5a)-(5b) and (6a)-

(6c), human capital accumulation (1), the goods market clearing condition (7), and the transversality

condition ˆlim 0.t

t

c tt

u b

From now on, prices and allocations with a superscript CE stand for those in competitive

equilibrium. Using (2a) and (2b), we rewrite the decisions of savings and education in (6a) and (6b),

respectively, as intertemporal and intergenerational consumption Euler equations,

1

ˆ

1

ˆ

( ),CEtt

CEt

c CE

t

d

uf k

u

= (8a)

1

ˆ

1

ˆ

( ) ( ).CEt

t

c CE CE

t t

c

ue w k

u n

(8b)

If condition (6c) is binding, it is rewritten as 1 1

ˆ ˆ.CE CE

t tnd c

u u

Substituting this relation into the

intertemporal consumption Euler equation (8a) gives ˆ

ˆ1

1( ).CEct

CEct

uCE

tu nf k

Then, using this equation and

(2b), we can rewrite the intergenerational consumption Euler equation (8b) as a relationship between next

period’s effective capital 1

CE

tk and this period’s education spending ,CE

te with the latter determining

next period’s human capital 1 .CE

th

8

1

1 1 1

( )( ) .

( ) ( )

CECE tt CE CE CE

t t t

f ke

f k k f k

(8c)

In steady state, the allocations in competitive equilibrium { , , , , ,, }CE CE CE CE CE CE CE

t t t t t t tkc d s e b h are

time-invariant, denoted by }{ , , , , ., ,CE CE CE CE CE CE CEc d k s e b h First, in the steady state, parent’s education

investment for children in (8b) is rewritten as

( ) ( ),CE CEn w k e (9a)

which gives a positive relationship between education costs eCE and effective capital CEk as follows.

1

( ) ,( )

CE CE

CE

ne e k

w k

( ) ( )( ) 0.

( ) ( )

CE CECE

CE CE

w k ee k

w k e

(9b)

Next, with the use of (1), (2a), (2b) and (9b), the budget constraints (4a) and (4b) enable us to write

consumption at middle age and when old as functions of effective capital and bequests as follows.11

( , ),CE CE CE CEc c k b (10a)

( , ),CE CE CE CEd d k b (10b)

where 0, 0, 0, 0.CE CE CE CE

CE CE CE CE

c c d d

k b k b

12

These two above relationships enable us to write effective consumption of middle-aged agents ˆCEc

and old agents ˆCEd as functions of CEk and CEb as follows.13

ˆ ˆ , ,CE CE CEc c k b ( ) (11a)

ˆ ˆ , .CE CE CEd d k b ( ) (11b)

Moreover, using (2a), we rewrite (8a) as ˆˆˆ ˆˆ ˆ( , ) ( ) ( , ) 0,CE CE CE CE CE

c du c d f k u c d and thus a zero net

marginal cost of savings (c.f. (6a)). Using (11a) and (11b), this condition can be expressed as a function

of CEk and CEb as follows.14

ˆˆˆˆ( , ) ( , ( ) ( ( , )) 0.CE CE CE CE CE CE CE

c dk b u c k b f k u d k b ( )) (11c)

11 That is, ( , ) ( ) ( ( )) ( ( )) ( )CE CE CE CE CE CE CE CE CEc k b w k e k b nk e k ne k and ( , ) ( ) ( ( )) .CE CE CE CE CE CE CEd k b f k nk e k nb

12 To derive these partial effects with respect to CEk , we have followed Alonso-Carrera et al. (2008) and assumed

that the negative effect of CEk onCEc dominates and the positive effect of CEk on

CEd dominates. Thus,

0CE

CE

c

k

and 0.

CE

CE

d

k

These conditions for 0

CE

CE

c

k

and 0

CE

CE

d

k

are easily satisfied, if the utility is logarithmic,

the production function is Cobb–Douglas, and the values of , ,c d c and d are small.

13 ˆ , ( , ) ( ( , ), ( , ))CE CE CE CE CE CE CE CE CE CE CEc k b c k b c k b d k b ( ) and ˆ , ( , ) ( ( , ), ( , )).CE CE CE CE CE CE CE CE CE CE CEd k b d k b c k b d k b ( )

14 Notice that ( , )CE CEk b is defined for CEk >0 and CEb >0 such that ˆ ,CE CEc k b( )>0 and ˆ ,CE CEd k b( )>0.

9

Differentiating ( , )CE CEk b yields 0CEb b

for all bCE>0 and 0CEk k

for all bCE0.15

Intuitively, a higher bequest increases middle-aged agents’ consumption, decreases old agents’

consumption and thus, decreases the net marginal cost of savings. By contrast, a higher effective capital

decreases middle-aged agents’ consumption, increases old agents’ consumption and thus, increases the

net marginal cost of savings.

It suffices to impose the same assumption made by Abel (1987) and Alonso-Carrera et al. (2008).

Assumption A. There exists a unique 0 0k satisfying ( ,0) 0CEk with ˆˆ ,0 0, ,0 0CE CEc k d k ( ) ( )

and 0( ,0) 0.

kk

This assumption posits that even with a zero bequest, there exists a positive effective capital CEk

consistent with positive consumption of middle-aged and old agents in the steady state. Moreover, even

with a zero bequest, ( ,0)CE

kk >0 means that ( ,0)CEk has a positive slope with respect to effective

capital. It is clear that properties 0CE

CE

c

k

and 0

CE

CE

d

k

ensure that Assumption A is met.

Note that Assumption A implies that ( ,0) 0CEk if 0CEk k and ( ,0) 0CEk if 0.CEk k

Yet, the bequest motive may or may not be operative and thus, bCE may be positive or zero.

To see when the bequest motive is operative or inoperative, we substitute the binding steady-state

condition for bequest choices (6c) ˆ ˆCE CEnd cu u

into the condition of a zero net marginal cost of savings

(11c), and then evaluate it at a zero bequest. According to Assumption A, then there exists a threshold

value of the altruism factor consistent with a zero bequest as follows.

0.

( )

n

f k

Thus, if agents’ altruism factor is larger than the threshold value , the bequest motive is

operative. Intuitively, when parents care for their children at a degree higher than the threshold value,

they leave bequests for children. By contrast, if is smaller than the threshold value , parents care

for their children at a degree less than the threshold value. Then, the bequest motive is inoperative and

parents do not leave bequests.16

15 In deriving these signs, the properties in (10a) and (10b) are used. 16 The threshold value of the altruism factor is similar to the one obtained in Weil (1987). Consumption externalities was introduced into Alonso-Carrera et al. (2008), which modified the threshold value of the altruism factor through their effects on physical capital per capita. Human capital investment is introduced here, which changes the threshold value of the altruism factor through its effect on physical and human capital; that is via the

10

Case 1. Operative bequest motives

Now, the altruism factor is larger than the threshold value , and thus bCE>0. The intertemporal

consumption Euler equation (8a) in the steady state, accompanied by (6c) with an equality, gives

( ).CEn f k (12a)

Then, in the steady state, a parent leaves bequests to the level, when the marginal cost of leaving

bequests to n children equals the altruism-factor discounted marginal product of effective capital resulting

from saving bequests for children. The condition implies a positive relationship between β and :CEk an

economy with a larger altruism factor leads to a larger effective capital per capita in the steady state.

Then, parents’ education investment for children in (9a), which is also the intergenerational

consumption Euler equation, accompanied by the use of (12a), yields

( ) ( ) ( ).CE CE CEw k e f k

This condition equalizes the effective wage for children and the marginal product of effective capital

resulting from saving bequests for children. This condition is also obtained in Bishnu (2013), but there

are differences. While agents borrow their own education costs in Bishnu (2013), here parents pay for

children’s education costs and have operative bequest motives. As a result, the marginal product of

effective capital is due to the gross loan interest payment in Bishnu (2013), but it is due to the decrease

in a parent’s savings resulting from saving bequests for children in our model. Note that, with the use of

(2b), the above condition is rewritten as

[ ( ) ( )] ( ) ( ),CE CE CE CE CEf k k f k e f k (12b)

and it is easy to show that 0.CE

CE

de

dk

Thus, when bequest motives are operative, (12a) and (12b) characterize steady-state effective capital

and education cost, and thus human capital, in competitive equilibrium.

Case 2. Inoperative bequest motives

In this case, the altruism factor is smaller than the threshold value. Then, the intertemporal

consumption Euler equation (8a) in the steady state, accompanied by (6c) with an inequality, gives

( ),CEn f k (13a)

in which a parent’s marginal cost from leaving bequests to n children is larger than the altruism-factor

discounted marginal product of effective capital resulting from saving bequests for children. Thus, a

effect on effective capital per capita 0k .

11

parent leaves no bequests to children, bCE=0.

Then, parents’ education investment for children in (9a) yields

( ) ( ) ( ).CE CE CEw k e f k

Now, with an inoperative bequest motive, the effective wage for children is larger than the marginal

product of effective capital resulting from saving bequests for children. Thus, parents pay for children’s

education but leave no bequests for children. Note that Bishnu (2013) does not get this condition.

With the use of (2b), the above condition is rewritten as

[ ( ) ( )] ( ) ( ).CE CE CE CE CEf k k f k e f k (13b)

2.2 Social planner’s allocations

This subsection studies the problem of a social planner (SP). A social planner takes into account

the consumption externalities overlooked by individuals. A planner’s resource constraint in period t is

1( ) .tt t t t t

dh f k c n e k

n (14)

A social planner maximizes the sum of lifetime utilities over generations subject to human capital

accumulation in (1) and the resource constraint in (14) for all t. As in Docquier et al. (2007) and Bishnu

(2013), the sum of lifetime utilities over generations is discounted by a factor λ ∈ (0, 1). Like these authors,

this discount factor is referred to as the social weight that a social planner attaches to future generations.

A high λ indicates a smaller social discount rate in that the social planner devalues less of future

generations. In the analysis that follows, a social planner with a different social weight λ may be

interpreted as a different social planner with each being indexed by her own λ. The Lagrangian of a social

planner’s problem is as follow:

1 1 1

0

ˆˆ( , ) ( ) ( ) ,t tt t t t t t t t

t

dL u c d q e f k c n e k

n

where λtqt is the multiplier associated with the economy’s resource constraint in period t.

The first-order conditions with respect to 1 0, , ,t t t t t

c d e k

are, respectively,

ˆˆ1 0,SP SP SP

tt t tc tc d c

u u q (15a)

ˆˆ1 0,SP SP SP

tt t t

tdc d d

qu u

n (15b)

1 1 1 1( ) ( ) ( ) 0,SP SP SP SP

t t t t t tq e f k f k k nq (15c)

1 1( ) 0,SP

t t tq f k nq (15d)

where superscript SP is used to stand for the planner’s outcome.

12

Definition 2. For a given social weight λ, and initial stock h0 and k0, the social optimum is the path

0

, , , ,SP SP SP SP SP

t t t t t tc d k e h

that satisfies the resource constraint (14), the first-order conditions (15a)-(15d),

and the transversality condition lim 0.t SP

t tt

q k

From (15a) and (15b), we obtain the social planner’s intratemporal consumption Euler equation.

ˆ

ˆ

1,

SPt

SPt

c

t

d

u

u (16)

where

1

1

d ct t

c dt t

n

t n

is the “externality factor,” which is time invariant, the reason being that σ and φ

both are linear, so σct, σdt, φct and φdt are time invariant. Hence, hereafter, σct, σdt, φct, φdt and Δt will be

denoted by σc, σd, φc, φd and Δ, respectively.

In (16), the social planner internalizes the consumption externality. If there is no externality, Δ＝n

and the social weight λ affects whether a social planner allocates more or less resources to future

generations. With consumption externalities, both the social weight and consumption externalities

influence whether a social planner would allocate more or less resources to future generations.

With the intratemporal consumption Euler equation (16), we can use (15a) and (15d) to derive the

social planner’s intertemporal and intergenerational consumption Euler equations, respectively, as follows.

1

ˆ

1

ˆ

( ),SPt

SPt

c SP

t

d

uf k

u n

= (17a)

1

ˆ

1

ˆ

( ).SPt

SPt

c SP

t

c

uf k

u n

(17b)

Moreover, we use (15c) and (15d) to obtain a different intergenerational consumption Euler

equation that relates next period’s effective capital 1

SP

tk to this period’s education spending ,SP

te which

determines next period’s human capital 1.

SP

th

1

1 1 1

( )( ) .

( ) ( )

SPSP tt SP SP SP

t t t

f ke

f k f k k

(17c)

In the steady state, (16) and (17a) are equal, which lead to ˆ

ˆ

1 ( ),SPc

SPd

uSP

u nf k

and thus

( ),SPn f k (18a)

which is the modified golden rule condition. This condition determines the golden rule effective capital

13

per capita .SPk Note that if a social planner assigns a larger social weight to future generations (a larger

), there would be a larger optimal effective capital per capita in the long run.

Finally, in a steady state, (17c) is

[ ( ) ( ) ] ( ) ( ),SP SP SP SP SPf k f k k e f k (18b)

and it is easy to show that 0.SP

SP

de

dk

Thus, (18a) and (18b) characterize steady-state effective capital and education cost in the social

optimum.

3. Competitive Equilibrium vs Social Optimum

This section studies whether or not physical and human capital in competitive equilibrium are equal

to what a social planner would have intended. The issue dates back Diamond (1965), who showed that

in the case of over-accumulation of physical capital, intergenerational rearrangement such as pensions is

valuable in increasing the welfare of all generations. However, the present context is more complicated,

as it encompasses not only human capital as a choice variable but also consumption externalities within

and across generations. From the previous section, it is clear that the planner’s choice of social weights

plays a central role in determining the optimal level of physical and human capital accumulation. There,

the planner solves a dynamic-planning problem with declining social weights over future generations. As

such, if we are to compare the allocations between the competitive equilibrium and the social optimum,

we need to find a way to make the laissez-faire allocation and the social optimum directly comparable.

The intertemporal and the intergenerational consumption Euler equations in the competitive

equilibrium are (8a) and (8c) and those in the social optimum are (17a) and (17c). While (8c) is the same

as (17c), (8a) is different from (17a). In comparing these two equations, we observe that their differences

lie in the effect of consumption externalities overlooked by individuals as summarized in the externality

factor Δ. These two optimality conditions coincide, only when there is no consumption externality and

thus, Δ=n. However, in the presence of consumption externalities, Δ≠n and thus, it does not guarantee

that CEk and SPk are identical. As such, although (8c) and (17c) look identical, the allocation of human

capital in the laissez-faire equilibrium is different from the social optimum, if CEk and SPk are different.

With consumption externalities and to make meaningful comparisons, we follow Bishnu (2013) and

establish a common point by devising the notion of a “laissez-faire supported” social weight, denoted by

. By construction, at this specific social weight, if there is no externality, the planner’s allocation

coincides with the laissez-faire allocation. However, in the presence of consumption externalities, at this

specific social weight, the laissez-faire allocation may differ from the planner’s allocation.

14

We begin our comparisons by contrasting competitive allocations with those preferred by utilitarian

planners at the laissez-faire supported social weight. Then, we proceed to comparisons with the social

weight different from this laissez-faire supported social weight. We focus on the comparisons in the

steady state. We start with the case when the bequest motive is operative.

3.1 Suboptimality of the equilibrium when the bequest motive is operative

In the steady state, competitive equilibrium conditions (8a) and (8c) become (12a) and (12b), and

the social optimum conditions (17a) and (17c) become (18a) and (18b). Note that, in comparing (12a)

and (18a), CEk is equal to or different from SPk when λ is equal to or different from β. Next, CEe in

condition (12b) and SPe in (18b) are the same or different, if CEk is equal to or different from .SPk

Denote by , ,i

j e j= k, h, i= CE, SP, the percentage change in physical capital and human capital

when the education expenditure is increased by one percent. That is, , /

i i

i i

i k ek e k e

and , / .

i i

i i

i h eh e h e

We can establish the following result.

Lemma 1. For i= CE, SP, , ,

i i

k e h e holds. For any , 0SPdk

d and 0.

SPde

d Moreover, if there exists a laissez-

faire supported social weight , it must be unique.

To show the first part, if we differentiate (12b) with respect to CEk and CEe for a given ,CEh and

also differentiate (12b) with respect to CEh and CEe for a given ,CEk then we obtain , , .CE CE

k e h e 17

Moreover, we differentiate (18b) to get , , .SP SP

k e h e Hence, , , ,i i

k e h e i= CE, SP. For the second part,

we differentiate (18a) to obtain 0.SPdk

d Moreover, differentiating (18b) yields 0

SP

SP

de

dk , implying 0.

SPde

d

Finally, to prove the third part, we rewrite SPk in (18a) as a function of λ: 1( ) ( ) ( ).SP SP nk k f

As λ∈(0, 1), then 0

lim ( ) 0,SPk

1

max1

lim ( ) ( ) ( )SP SPk f n k

and thus, ( )SPk max(0, ).SPk Suppose

that there exists a laissez-faire supported social weight . Since SPk strictly increases in , it follows

that there exists a unique level of SPk that corresponds to . If we denote by ( )SPk the level of

SPk corresponding to , then max( ) (0, ).SP SPk k

In words, the first part says that if the education expenditure increases by one percent, the

17 Differentiating (12b) yields ( )ln ln

, ,ln ln ( )0,

CE CECE CE

CE CE CE

e f k fCE CEd k d hk e h ed e d e k f w

as 0,CEf k f w 0, and

0.f w As a result, , , .CE CE

k e h e

15

proportional increase of physical capital per capita is larger than the proportional increase of human

capital per capita. The second part says that both effective capital per capita and education investment in

the social optimum increase in the social weight. Finally, the last part says that, given a social optimum

and thus the level ,SPk there is a unique for each possible Δ, where depends on Δ.

Other than the uniqueness result of ( )SPk , we also establish a uniqueness result.

Lemma 2. For any and under operative bequest motives, if there exists a * such that *( )SP CEk k holds,

then *( ) ,SP CEe e and vice versa.

The uniqueness result in Lemma 2 follows directly from (12b) and (18b), because there is no

spillover from human capital and the form of (12b) for CE is the same as that of (18b) for SP.

We are ready to compare the allocations in CE with those in SP. As in Bishnu (2013), we will focus

on ,k the ratio of physical capital k to human capital h. Based on the results of Lemma 2, one can easily

verify that given feasibility, h and k in CE and SP coincide at a unique social weight. According to the

uniqueness results in Lemma 2, we start our analysis with the special laissez-faire supported social weight

. This specifies the weight at which the allocations in CE and SP are identical in an economy without

consumption externalities. However, with consumption externalities, although the marginal rate of

substitution in CE is still the same as in SP at , the accumulated levels of h and k in CE are different

from the SP at this specific social weight. Thus, the above result no longer remains valid.

In the rest of the analysis, we proceed by varying the value of λ until we arrive at that specific social

weight, where the levels of h and k in SP are identical to those in CE. For simplicity, we follow Bishnu

(2013) and assume no population growth from now on; thus n=1. Then, Δ> (resp. = or <) n=1 if

.( ) ( ) ( );c c d dresp or that is, if the consumption externality from the middle-aged

dominates (resp. equals or is dominated by) that from the old, then Δ> (resp. = or <) 1.

Case 1. The social weight is equal to the laissez-faire supported social weight

Starting with the case of , if we compare (12a) in CE with (17a) in SP, it is clear to see that

(12a) is different from (17a) when Δ≠1. First, when Δ>1, it is obvious that ( ) ,SP CEk k which implies

( )SP CEe e and thus, ( ) ,SP CEh h according to Lemma 1. Moreover, the results of ( )SP CEk k

and ( )SP CEh h imply ( ) .SP CEk k Next, when Δ<1, it is easy to see that ( )SP CEk k ,

( )SP CEh h and ( ) .SP CEk k Thus, the allocation in CE differs from that in SP in the same direction,

16

with under-accumulating human and physical capital in the CE when Δ>1 and over-accumulating human

and physical capital in CE when Δ<1.

Case 2. The social weight is different from the laissez-faire supported social weight

Now, we proceed to the case . First, when Δ<1, according to Lemma 1, wherein 0SPdk

d

and 0SPde

d and thus, 0

SPdh

d , there exist

1 and 2 such that 1( )SP CEk k and

2( ) .SP CEh h Besides, according to Lemma 2, we can get 1 2 , such that ( )SP CEk k

holds. It follows that, for all , we have SP CEh h and ;SP CEk k and for all , we have

SP CEh h and .SP CEk k Next, when Δ>1, it is easy to show that there exists such that

( )SP CEh h , ( )SP CEk k and ( )SP CEk k hold. It follows that, for all , we have

SP CEh h and ;SP CEk k and for all , we have SP CEh h and .SP CEk k

Thus, when the social weight λ differs from the laissez-faire supported social weight , the

allocation in CE also differs from the SP in the same direction, with over-accumulating human and

physical capital in CE when Δ<1, and under-accumulating human and physical capital in CE when Δ>1.

To summarize these results, we have the following proposition.

Proposition 1. At any λ and the presence of consumption externalities, when the bequest motive is operative, the levels

of physical and human capital in competitive equilibrium are different from those in the social optimum in the same directions.

Thus, when the bequest motive is operative, no matter whether the social weight is equal to the

laissez-faire supported social weight or not . ., ,( )i e or the accumulation levels of physical

capital and human capital in competitive equilibrium can never be different from the planner’s choice in

opposite directions. Figure 1 illustrates physical and human capital in competitive equilibrium and the

social optimum under different ranges of social weights for different values of the externality factor. It

is clear to see that solid green line and the dashed red line for physical capital k and human capital h does

not overlap each other for all different values of Δ.

[Insert Figure 1 here.]

3.2 Suboptimality of the equilibrium when the bequest motive is inoperative

This subsection analyzes the situation when the bequest motive is inoperative. In this environment,

(6c) is not binding and thus, b=0. The CE in the steady state is characterized by (13a) and (13b), which is

compared with (18a) and (18b) of the SP in the steady state. As we will see, the accumulation levels of

17

physical and human capital in CE can be different from the SP in opposite directions for some values of

the social weight.

First, note that without consumption externality (Δ=1), ( ) ( )SP CEk k at the laissez-faire

supported social weight. Then, by comparing (13a) with (18a), it is clear that .

Now, the bequest motive is inoperative. Lemma 1 remains valid. To see this, if we compare a parent’s

education investment condition in (9a) in CE with (18b) in SP, it is easy to verify that for all t, 0i

i

k

e

, i=

CE, SP, and thus , ,

i i

k e h e holds in the steady state. Moreover, it is easy to show that if a laissez-faire

supported social weight exists, it must be unique. Therefore, Lemma 1 holds true here.

However, Lemma 2 no longer holds. If we compare the CE condition (13b) with the SP condition

(18b), the equality results of *( )SP CEk k and *( )SP CEe e in Lemma 2 do not hold. Given that

( ) 0,e from the comparison of (13b) with (18b), one can easily establish the following lemma.

Lemma 3. For any , if there exists a 1̂ such that 1ˆ( )SP CEk k holds, then 1

ˆ( ) ;SP CEe e and if there exists a

2̂ such that 2ˆ( )SP CEe e holds, then 2

ˆ( ) .SP CEk k In addition, 1 2? .

Now, as the bequest motive is inoperative, there exist 1 2? , such that 2

ˆ( )SP CEk k and

1ˆ( ) .SP CEe e That is, under an inoperative bequest motive, the accumulation levels of physical and

human capital in CE may be different from those in SP in different directions. To see this, we first start

with the case , followed by the case .

Case 1. The social weight is equal to the laissez-faire supported social weight

With , first, when Δ>1, we have known ( )SP CEh h , ( )SP CEk k and ( ) .SP CEk k

Thus, like Proposition 1, the levels of physical and human capital in CE are different from those in SP in

the same directions. Next, when Δ=1, we obtain ( )SP CEk k which, according to Lemma 3, implies

( ) ,SP CEe e and thus ( ) .SP CEh h With ( )SP CEk k and ( ) ,SP CEh h it follows that

( )SP CEk k . Thus, when Δ≥1, like Proposition 1, physical capital and human capital in CE are both

under-accumulated as compared to the SP.

However, when Δ<1, then ( )SP CEk k does not imply ( ) ,SP CEe e according to Lemma 3.

Thus, it may lead to ( )SP CEh h or ( ) .SP CEh h Hence, it is not sure of whether human capital in

18

CE is under- or over-accumulated as compared to the SP. For a similar reason, it is not sure of whether

physical capital in CE is under- or over-accumulated as compared to the SP. We summarize one of our

two main findings as follows.

Proposition 2. With an inoperative bequest motive and the social weight equal to the laissez-faire supported social weight

, if the consumption externality from the old dominates that from the middle-aged, the accumulation levels of physical

and human capital in competitive equilibrium may differ from the social optimum in opposite directions.

Case 2. The social weight is different from the laissez-faire supported social weight

Now, we proceed to the case . First, when Δ<1, according to Lemma 1, ( )SP CEk k and

( ) .SP CEh h Hence, there exist 1 and

2 such that 1( )SP CEk k and

2( ) .SP CEh h

Moreover, according to Lemma 3, we know 1( )SP CEh h and

2( ) ,SP CEk k which imply

1 2( ) ( ).SP CE SPk k k Since SPk is continuous in , there exists 3 2 1( , ) such that

3( ) .SP CEk k As a result, we obtain (1) CE SPh h and CE SPk k for all3 , and (2) CE SPh h

and CE SPk k for all 2 , but (3) CE SPh h and CE SPk k for all

2 3( , ).

Similarly, when Δ>1, we follow the above procedure and can easily show that there exist 2 and

3 with 2 3 such that (1) SP CEh h and SP CEk k for all

3 , and (2) SP CEh h

and SP CEk k for all 2 , but (3) SP CEh h and SP CEk k for all

2 3( , ). .

Finally, when Δ=1, one can easily show that the regimes of the social weight separating under- or

over-accumulation of physical and human capital are like those when Δ>1, except now 2 3 .

Thus, we obtain (1) SP CEh h and SP CEk k for all 3 , and (2) SP CEh h and SP CEk k for

all 2 , but (3) SP CEh h and SP CEk k for all

2 3( , ).

Based on the above analysis, we construct the following corollary.

Corollary 1. Assume that the bequest motive is inoperative.

(i) If the externality factor is greater than one, i.e., >1, there exist social weights 2 3 such that, for any

3( . ) ,resp the CE over-accumulates (resp. under-accumulates) physical capital, but for any

2( . ) ,resp the CE over-accumulates (resp. under-accumulates) human capital.

(ii) If the externality factor is equal to one, i.e., =1, there exist social weights 2 3 such that, for any

19

3( . ) ,resp the CE over-accumulates (resp. under-accumulates) physical capital, but for any

2( . ) ,resp the CE over-accumulates (resp. under-accumulates) human capital.

(iii) If the externality factor is less than one, i.e., <1, there exist social weights 3 2 such that, for any

3( . ) ,resp the CE under-accumulates (resp. over-accumulates) physical capital, but for any

2( . )resp , the CE under-accumulates (resp. over-accumulates) human capital.

Figure 2 illustrates the range for the values of social weights under different values of the externality

factor Δ wherein physical and human capital in CE differ from the SP in opposite directions. It is clear

to see that, under the different values of Δ, there exists a range of moderate values of social weights such

that the solid green line and the dashed red line for capital k and human capital h overlap each other. In

particular, in CE, capital over-accumulates and human capital under-accumulates.

[Insert Figure 2 here.]

We summarize another main result in the following proposition.

Proposition 3. With an inoperative bequest motive, for an externality factor that is larger than (resp. equal to, or smaller

than) one, there exists a corresponding range of social weights different from the laissez-faire supported social weight, such

that the accumulation levels of physical and human capital in CE are different from the SP in opposite directions.

Thus, Propositions 2 and 3 together indicate that when the bequest motive is inoperative, for an

externality factor, there exists values of the social weight, such that the accumulation levels of the two

types of capital in competitive equilibrium differ from the planner’s allocations in opposite directions.

4. Optimal Policy

In this section, we characterize the optimal policy in order to implement the SP allocation. We

consider a tax policy consisting of an estate tax, a capital income tax, an education subsidy and a system

of lump-sum taxes and subsidies. We follow Section 3 and maintain no population growth, and thus n=1.

Concerning the lump-sum taxes, we assume that working-aged agents in period t pay a lump-sum

tax 0.m

tT The revenue is devoted to financing a lump-sum transfer to the old 0o

t in period t.

Thus, one of the government constraints is

0.m o

t tT (19a)

Next, working-aged agents also pay an estate tax rate b

t on the inheritance they receive in period

t and old agents pay a capital income tax rate k

t on returns to savings in period t. These tax revenues

20

are used to subsidize period-t education costs .e

t te Thus, another government budget constraint is

1 .b k e

t t t t t t tb R s e (19b)

The budget constraints faced by an individual of generation t during the middle age of life in (4a)

and the old age of life in (4b) are now modified, respectively, as

(1 ) (1 ) ,m b e

t t t t t t t t tw h T b c s e (20a)

1 1 1 1 1(1 ) .k o

t t t t t tR s d b (20b)

The optimization problem of the representative agent of generation t is to maximize (3) with respect

to 1 1 0, , ,t t t t t

c d e b

subject to (1), (20a) and (20b), given wt and Rt+1 for all t and given k0 and h0. The

optimality conditions for st, et and bt+1 are as follows.

1ˆˆ 1 1 1 1

ˆ ˆˆ ˆ( , ) ( , )(1 ) ,t t

k

c t t t t t tdu c d u c d R

(21a)

1ˆ ˆ1 1 2 1

ˆˆ ˆˆ( 1 ) ( , ) ( , ) ( ) ,t t

e

t c t t c t t t tu c d u c d w e

(21b)

11

ˆ ˆ1 1 2 1ˆˆ( , ) ( ,ˆ )(1 ),ˆ

tt

b

t t c t t tdu c d u c d

(21c)

where the equality in (21c) holds if bt+1>0.

Definition 3. For given h0 and k0 and a given path of tax policies 0{ , , , , } ,m o b k e

t t t t t tT

a competitive

equilibrium is a path 0{ , , , , , , , , }t t t t t t t t t tc d k s e b h w R

that satisfies firms’ optimization (2a)-(2b), agents’

optimization (21a)-(21c), the human capital accumulation (1), the goods market clearing condition (7),

the government budget constraints (19a)-(19b), and the transversality condition ˆlim 0.t

t

c tt

u b

Let a value and a policy with a “crescent” symbol on their top as an optimal value and an optimal

policy in competitive equilibrium. Then, the path of the optimal policy is 0

, , , ,m o b k e

t t t t tt

T

. The

optimal policy is such that the tax-corrected equilibrium path of , , ,t t t tc d k e coincides with the optimal

path of , , , .SP SP SP SP

t t t tc d k e By substituting savings from the budget constraint of the old in (4b) into

the goods market condition in (7), the optimal path of bequest implied by the planner’s solution is

( ) .SP SP SP SP SP

t t t t tb f k k h d (22)

If the optimal amount of bequest is positive, this implies that the bequest motive is operative along

an equilibrium path. By contrast, a negative optimal amount of bequest implies that the bequest motive

is inoperative along this equilibrium path. Then, mimicking this optimal path of bequest along the

equilibrium path implies that the equilibrium bequest path attains the first best solution. To characterize

21

the optimal tax policy, we distinguish two cases.

4.1 When the bequest motive is operative

In this case, the equilibrium path associated to the optimal policy is characterized by (14), (20a)-

(20b) and (21a)-(21c). Note that, along with the use of (2a) and (2b), (21a)-(21c) lead to, respectively,

1

ˆ

1 1

ˆ

(1 ) ( ),t

t

c k

t t

d

uf k

u

(23a)

1

ˆ

1 1 1

ˆ

1( ) ( ) ( ),

1

t

t

c

t t t te

c t

uf k k f k e

u

(23b)

ˆ

ˆ

1.

(1 )

t

t

c

b

td

u

u

(23c)

To investigate which policy instruments should be used in order to achieve a social optimum in a

market economy, we compare social optimum conditions (16), (17a) and (17b) with tax-corrected

competitive equilibrium conditions (23c), (23a) and (23b), respectively, along with the use of (17c). We

obtain the following result.

Proposition 4. When the bequest motive is operative, the optimal taxes are 1 ,=k

t 1 ,e

t

and

11 .b

t

When the bequest motive is operative, consumption externalities are the only source of

suboptimality. In this case, consumption externalities and altruism affect decisions on savings, education,

and bequests. Thus, the optimal policy depends on consumption externalities and altruism. Yet, lump-

sum taxes are irrelevant due to Ricardian equivalence.

First, the sign of optimal capital taxes k

t depends only on the externality factor Δ. The optimal

capital tax is positive if the externality factor is smaller than unity,18 but is negative if otherwise. Intuitively,

if Δ<1 and thus, the consumption externality coming from the working-aged is weaker than that coming

from the old, the working-aged save too much and consume too little. Then, a positive capital tax serves

to lower their savings. By contrast, if the consumption externality coming from the old is stronger, the

working-aged save too little and consume too much. Then, a subsidy to capital serves to increase the

savings of the working-aged.

Next, the education subsidy e

t is not affected by consumption externalities but is affected by the

18 Namely, if the externality factor Δ > (resp. < ) 1; i.e., if ( . .)c c d dresp

22

altruism motive. The education subsidy is positive if the altruism factor β is smaller than the social weight

λ, and is negative if otherwise. Intuitively, education is a parent’s transfer in kind to children. If the

altruism factor is smaller than the social weight, parental education investment for children is less than

what arranged by the social planer. Hence, an education subsidy is optimal. Conversely, if the altruism

factor is larger than the social weight, parental education investment for children is more than what

devised by the social planer. Thus, an education tax is optimal.

Finally, a bequest is a mix of savings and transfer in goods. The sign of the bequest tax b

t depends

not only on consumption externalities Δ but also on the altruism motive. If Δ=1, the effects of

consumption externalities from the working-aged completely offset those from the old. Then, only the

altruism (i.e., transfer) motive is at work, and the altruism factor relative to the social weight is what

matters for the sign of a bequest tax. In this case, if the altruism factor β is smaller than the social weight

λ, the old leave bequests less than the optimum, and a subsidy to bequests is optimal; if otherwise, a tax

on bequests is optimal. Alternatively, if Δ≠1, consumption externalities are also at work. Then, if the

altruism-factor discounted externality factor βΔ is larger than the social weight λ, the old leave bequests

more than the optimum, and a bequest tax is optimal; if otherwise, a subsidy to bequests is optimal.

4.2 When the bequest motive is inoperative

In this case, (21c) is not binding and bt=0. The equilibrium path associated to the optimal policy is

characterized by (14), (20a)-(20b), (21a)-(21b), and bt=0. The optimal policy is then analyzed by

comparing (17a) and (17b) with (21a) and (21b), respectively. With the use of (17c), we get the optimal

capital tax rate 1=k

t and the education subsidy rate 1 ,e

t

which are the same as in the case

when the bequest motive is operative.

The suboptimality due to the inoperativeness of the bequest motive is resolved through lump-sum

taxes. To obtain the optimal taxes due to inoperative bequest motive, we substitute the period t

government budget constraint in (19b) into the period t old-age individual’s budget constraint in (20b),

with the use of the rate of returns ( )t tR f k in (2a) and the capital accumulation 1t t t ts k k h in

(7), along with n=1 and bt=0. Then, we obtain consumption of the old in period t as follows.

( ) .e o

t t t t t t td f k k h e (24)

In order for consumption, capital and human capital in the tax-corrected market equilibrium

condition in (24) to replicate the social optimum in (22),19 the lump-sum transfer to the old in period t

19 Note that (22) can be rewritten as ( ) .SP SP SP SP SP

t t t t td f k k h b

23

is set at .o e SP

t t t te b By using (19a), it follows that the lump-sum transfer to the old is paid by the

lump-sum tax to the working-aged in period t and thus, .m e SP

t t t tT e b The optimal taxes are as follows.

Proposition 5. When the bequest motive is inoperative, the optimal taxes are 1 ,=k

t 1 ,e

t

and

.o m e SP

t t t t tT e b

Thus, even though the bequest motive is inoperative, capital taxes and education subsidies are the

same as those in the case when the bequest motive is operative, because consumption externalities and

transfer motives remain at work. Now, the amount of bequest is zero in equilibrium and thus, a bequest

tax or a bequest subsidy is not feasible. However, the optimal bequest desired by the social planner in (22)

is negative, 0SP

tb . In this case, the social optimum calls for intergenerational lump-sum transfers.

Note that 0SP

tb indicates inoperative bequest motives, wherein the altruism factor β is less than

the threshold . First, if the altruism factor β is smaller than the social weight λ, 0e

t and thus,

education is subsidized at the amount 0.e

t te In this case, it is optimal to subsidize retirees a lump

sum equal to the difference between the education subsidy and the optimal bequest 0o e SP

t t t te b

and to tax the working-aged the same lump sum. Alternatively, if the altruism factor β is larger than the

social weight λ, 0e

t and thus, education is taxed at the amount 0.e

t te In this case, it is still

optimal to subsidize retirees a lump sum 0o e SP

t t t te b and tax the working-aged the same lump

sum if the absolute value of optimal bequests desired by the social planner SP

tb is larger than the

amount of education taxes ,e

t te and it is optimal to tax retirees a lump sum and subsidize the working-

aged the same lump sum if otherwise.

5. Concluding Remarks

This paper studies suboptimality of physical and human capital accumulation in a three-period OLG

model with consumption externalities and altruism. In the model, parents have willingness to finance

education and leave bequests for children.

We find that parents are always willing to finance children’s education, but the motive to leave

bequests may be operative or inoperative. The mechanism leading to suboptimality in our model depends

on consumption externalities and altruism. When the bequest motive is operative, the lack of optimality

is due to consumption externalities. Then, like the existing model with consumption externalities but

without parental altruism, there is either over-accumulation or under-accumulation of both physical and

24

human capitals. However, when the bequest motive is inoperative, suboptimality is due to consumption

externalities and inoperative bequest motives. In this situation, for a range in the value of the social weight,

physical capital is over-accumulated but human capital is under-accumulated in equilibrium, which lends

support to the policy of implementing education subsidies and public pensions at the same time, a policy

prevailing in OECD countries.

Optimal tax policies are examined in order for the equilibrium to attain the first best solution. Capital

is taxed or subsidized, depending on the externality factor being smaller or larger than unity, and

education is subsidized or taxed depending on the altruism factor being smaller or larger than the social

weight. Only under operative bequest motives, bequests are taxed or subsidized, depending upon the

altruism-factor discounted externality factor being larger or smaller than the social weight. By contrast, if

bequest motives are inoperative, there is no bequest in equilibrium, but the optimal bequest is negative.

Then, the social optimum calls for intergenerational lump-sum transfers from the working-aged to

retirees when the education is subsidized or taxed at an amount smaller than the absolute value of optimal

bequests, and calls for intergenerational lump-sum transfers from the retirees to the working-aged if

otherwise.

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27

Note: OA: Over-accumulation; UA: Under-accumulation.

Figure 1. Over- or under-accumulation of both types of capital when the bequest motive is operative.

.

Note: OA: Over-accumulation; UA: Under-accumulation.

Figure 2. Over- or under-accumulation of both types of capital when the bequest motive is inoperative.

k

k

k

Δ = 1

Δ < 1

𝜆

𝜆

0 𝜆− 𝜆

1

0 𝜆 = 𝛽 1

0 𝜆+ 𝜆 1

Δ > 1

𝜆

OA

UA

h

h

h

𝜆+1 𝜆+

3 𝜆+2

𝜆−2

𝜆−2

k

k

k

Δ = 1

Δ < 1

𝜆

𝜆

0 𝜆−3

𝜆

1

0 𝜆−3 𝛽 𝜆 1

0 𝜆 1

Δ > 1

𝜆

OA

UA

h

h

h